Study of abstract operator equations and their applications, specifically to point heat source control problems
otelbaevm@mail.ru
GOAL
Project Goal
The project has two main goals:
- to develop a function-theoretic approach to studying the solvability of abstract parabolic equations;
- to create a prescribed thermal field in a three-dimensional domain and an algorithm for controlling the solution of this problem using a laser heat source.
ABOUT THE PROJECT
Abstract
In this project, the original initial-boundary value problem for differential equations will be reduced to equivalent equations with bounded operators; the solvability of the main initial-boundary value problem for an abstract Navier–Stokes-type equation in function spaces will be studied; the solvability of the main problem with finite operators and their use in strong solutions will be investigated; and an algorithm for controlling a point heat source will be presented. The classical abstract Navier–Stokes-type boundary value problem covers a wide range of topics related to the generality of the equation, the boundary conditions or the domains in which these equations are considered. These problems model many phenomena studied in mechanics and physics. The project envisages intensive research in the area of local parabolic initial-boundary value problems.
Project Objectives
Main objectives of the project: to achieve the first goal, the solvability of abstract parabolic equations in a Hilbert space is studied; for the second goal, the final problem of controlling a point (laser!) heat source is formulated and solved. The main hypothesis of the project is that the methods and theories developed for the direct Navier–Stokes problems for homogeneous and inhomogeneous fluids are well-founded.
RESULTS
Scientific Achievements
It is known that living cells die when they find themselves in an environment with a temperature above a certain t0. This property is also characteristic of malignant cancer cells, which, as a rule, die faster than healthy cells of the body, and this fact is used in medicine. To harm healthy cells as little as possible, it is desirable to create a higher temperature in the vicinity of the malignant tumor while keeping the temperature in other parts of the patient's body close to the normal body temperature of a healthy person. Therefore, there is a need to create a controlled thermal field in the patient's body. In doing so, it is of course necessary to take into account a) the inaccessibility of the interior of the affected areas of the patient's body;
b) the non-uniform heat-conducting properties of the body (a variable thermal conductivity coefficient), as well as heat transfer by blood. It is well known in science that the heat equation is used to describe changes in the thermal field, and for an even more accurate description, hydrodynamic equations with variable mass are added (a special case of which is the classical Navier–Stokes equation). The simplest use of a thermal field is a method in which the patient is immersed in a hot bath with only the head above the surface. According to doctors in Novosibirsk, "this has an effect, and some cancer patients have been cured". Perhaps they are right and "cancer patients can sometimes be cured", but there may be cases when the patient is "boiled". Today, waves are applied to areas of the patient's body surface. This actually heats in the same way as a microwave oven. This method is more effective, to some extent ensures the locality of heating and rules out "boiling" the patient. In this case, however, heating is applied from the surface of the body, and the equations for the thermal field obey the classical maximum principle — such a method cannot get around the maximum principle. The maximum principle has been proven as a mathematical theorem and plays an essential role in the equations of mathematical physics; it cannot be circumvented by thermal action on the surface. For this reason, the volume of healthy body areas where the temperature is higher than in the diseased area (the cancerous tumor) may be of the same order as the volume of the tumor. It follows that too many healthy cells would be killed.
We propose I) to use a controlled point heat source on the surface of the patient's body; II) to circumvent the maximum principle, to use one of the variants of needles (devised by us). In this project, we show that by controlling a point heat source on the surface, as well as special needles, it is possible to obtain any predefined thermal field in an area of the body with any predefined accuracy. The results have been published in the following works by the project team: B.E. Kanguzhin, B.D. Koshanov, Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation l(·) − A with the Tricomi Operator A // Differential Equations, 2023, vol. 59, no. 1, pp. 4–14 (Kanguzhin, B.E., Koshanov, B.D. Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation l(·)−A with the Tricomi Operator A. Diff Equat 59, 1–12 (2023). https://doi.org/10.1134/S0012266123010019); Aitzhanov, S.; Bekenayeva, K.; Abdikalikova, Z. Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator. Mathematics 2023, 11, 3987. https://doi.org/10.3390/math11183987 (Scopus percentile 88). A test bench demonstrating the thrust force generated by oscillatory motion from a drive based on electromagnetic elements was designed and built, and tests were carried out.
INTELLECTUAL PROPERTY
Legal Protection
All results are published in open sources and do not require patenting.
IMPLEMENTATION
Commercialization and Practical Application
Research on the topic is mainly theoretical and fundamental. The results can be applied to modeling technological processes using advanced modern results of the theory of differential operators and to creating new original methods of research and analysis. This project is a direct continuation of the authors' research under program AP08857604 (Boundary and inverse problems for Navier–Stokes equations of homogeneous, inhomogeneous fluids, thermal convection and Kelvin–Voigt fluids, 2020–2022), funded by a grant of the Ministry of Education and Science of the Republic of Kazakhstan.
PUBLICATIONS
List of Publications
2023
- B.E. Kanguzhin, B.D. Koshanov, Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation l(·) − A with the Tricomi Operator A // Differential Equations, 2023, vol. 59, no. 1, pp. 4–14 (in Russian) (Kanguzhin, B.E., Koshanov, B.D. Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation l(⋅)−A with the Tricomi Operator A. Differential Equations 59, 1–12 (2023). https://doi.org/10.1134/S0012266123010019 ) (Scopus percentile 46, Web of Science Q2)
- Aitzhanov, S.; Bekenayeva, K.; Abdikalikova, Z. Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator. Mathematics 2023, 11, 3987. https://doi.org/10.3390/math11183987 (Scopus percentile 90, Web of Science Q1)
- B. Koshanov, M. Bakytbek, On Fredholm Solvability and on the Index of the Generalized Neumann Problem for an Elliptic Equation of High Order on a Plane // Extended Abstracts MWCAPDE 2023, https://doi.org/10.1007/978-3-031-41665-1, 167-176 pp.
- Zhusupova D., Abdikalikova Z., Ydyrys A., Burgumbayeva S., Modeling the operation of a gas compressor station to minimize fuel costs for surge protection // Conference "Modern Problems of Differential Equations and Their Applications", Tashkent, 2023, pp. 159–161
- Bekenayeva K.S., Aitzhanov S.E., Solvability of a pseudoparabolic equation with a fractional Caputo derivative // Conference "Modern Problems of Differential Equations and Their Applications", Tashkent, 2023, pp. 169–170
- Koshanov B.D., Sabirzhanov M.T., On some nonlocal boundary value problems for second order elliptic systems // Conference "Modern Problems of Differential Equations and Their Applications", Tashkent, 2023, pp. 181–184
- Otelbayev M., Koshanov B.D., Zhusupova D.S., Kozhobekova P.Zh., On a point heat source control problem and its application in medicine // Conference "Modern Problems of Differential Equations and Their Applications", Tashkent, 2023, pp. 248–249
- Otelbayev M., Kozhobekova P., Creating a thermal field using electromagnetic fields // Conference "Modern Problems of Differential Equations and Their Applications", Tashkent, 2023, pp. 105–106
2024
- Kanguzhin, B.; Koshanov, B., Criteria for the Uniqueness of a Solution to a Differential-Operator Equation with Non-Degenerate Conditions // Symmetry 2024, 16, 210, https://doi.org/10.3390/sym16020210 (Scopus percentile 94, Web of Science Q2)
- Dinara Zhussupova, Mukhtarbay Otelbaev, Saule Burgumbayeva, Modeling Gas Compressor Station Operation to Minimize Fuel Costs for Surge Zone Protection // International Journal of Rotating Machinery, Volume 2024, Article ID 5560308, 18 pages, https://doi.org/10.1155/2024/5560308 (Scopus percentile 46, Web of Science Q4)
- S.Burgumbayeva, D.Zhussupova, B.Koshkarova, Mathematical modelling of the process of natural gas transportation via pipe networks using crossing-branch method // JMMCS №1(121), 2024, DOI: https://doi.org/10.26577/JMMCS202412112 (CQASHE)
- S.E. Aitzhanov, A.A. Issakhov, A.S. Kassymbekova, Z.Abdikalikova, Solvability of inverse problem of a pseudoparabolic equation with fractional Caputo derivative // Solvability of inverse problem of a pseudoparabolic equation with fractional Caputo derivative . Journal of Mathematics, Mechanics and Computer Science, 2024, 124(4), 3–25. https://doi.org/10.26577/JMMCS2024-v124-i4-a1 (CQASHE)
- A. Ydyrys, Real-Time Monitoring of Athletes’ Heart Conditions During Intensive Fitness Using a Wearable Device with Abstract Operator Equations // Retos, in press (Scopus percentile 68, Web of Science Q3)
- Koshanov B.D., Sultangaziyeva Zh.B., On the well-posedness of nonlocal boundary value problems with an integral condition for higher-order quasi-hyperbolic equations // Conference "Non-classical Equations of Mathematical Physics", Tashkent, 2024, p. 178
- B. Koshanov, N. Oralbekova, Green’s functions of some boundary value problems for polyharmonic operators and their correct narrowings // Conference "Non-classical Equations of Mathematical Physics", Tashkent, 2024, p. 85
TEAM
Research Team
Otelbayev M., Dr. Phys.-Math. Sci., Professor
IITU JSC, Professor
Scopus ID: 6508334429
Researcher ID: A-2864-2015
ORCID: 2864-2015
Researcher ID in Publons: A-2864-2015
Koshanov B.D.
Chief Researcher
Scopus ID: 24331898200
Researcher ID: AAQ-7897-2020
ORCID: 0000-0002-0784-5183
Aitzhanov S.E.
Researcher
Scopus ID: 56656636600
ORCID: 0000-0001-5877-7195
Abdikalikova Z.T.
IITU JSC, Assistant Professor
Scopus ID: 37035907100
ResearchGate
Ydyrys A.Zh.
IITU JSC, Assistant Professor
Scopus ID: 57208477171
ResearchGate
Sultangaziyeva Zh.B.
Researcher
Scopus ID: 57221665566
Zhunusova D.S.
Researcher
Scopus ID: 57211348025
Researcher ID: EIY-4533-2022
ORCID: 0000-0002-1860-0843
Tokmukhamedova F.K.
IITU JSC, Senior Lecturer
Scopus ID: 57250078700
Researcher ID: 4870487
ORCID: 0000-0001-7980-8245
Sarsenbayev N.S.
Engineer
Nazar A.N.
Engineer
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